Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{x - 1} + \frac{x}{x + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))

C

double code(double x) {
	return (1.0 / (x - 1.0)) + (x / (x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x - 1.0d0)) + (x / (x + 1.0d0))
end function

Java

public static double code(double x) {
	return (1.0 / (x - 1.0)) + (x / (x + 1.0));
}

Python

def code(x):
	return (1.0 / (x - 1.0)) + (x / (x + 1.0))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x - 1.0)) + Float64(x / Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x - 1.0)) + (x / (x + 1.0));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x - 1} + \frac{x}{x + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))

C

double code(double x) {
	return (1.0 / (x - 1.0)) + (x / (x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x - 1.0d0)) + (x / (x + 1.0d0))
end function

Java

public static double code(double x) {
	return (1.0 / (x - 1.0)) + (x / (x + 1.0));
}

Python

def code(x):
	return (1.0 / (x - 1.0)) + (x / (x + 1.0))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x - 1.0)) + Float64(x / Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x - 1.0)) + (x / (x + 1.0));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + x} + \frac{1}{x + -1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ x (+ 1.0 x)) (/ 1.0 (+ x -1.0))))

C

double code(double x) {
	return (x / (1.0 + x)) + (1.0 / (x + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / (1.0d0 + x)) + (1.0d0 / (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	return (x / (1.0 + x)) + (1.0 / (x + -1.0));
}

Python

def code(x):
	return (x / (1.0 + x)) + (1.0 / (x + -1.0))

Julia

function code(x)
	return Float64(Float64(x / Float64(1.0 + x)) + Float64(1.0 / Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (x / (1.0 + x)) + (1.0 / (x + -1.0));
end

Wolfram

code[x_] := N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{x}{1 + x} + \frac{1}{x + -1} \]
4. Add Preprocessing

Alternative 2: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.75 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.75) (not (<= x 1.6)))
   (/ x (+ x (/ -1.0 x)))
   (+ x (/ 1.0 (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.75) || !(x <= 1.6)) {
		tmp = x / (x + (-1.0 / x));
	} else {
		tmp = x + (1.0 / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.75d0)) .or. (.not. (x <= 1.6d0))) then
        tmp = x / (x + ((-1.0d0) / x))
    else
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.75) || !(x <= 1.6)) {
		tmp = x / (x + (-1.0 / x));
	} else {
		tmp = x + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.75) or not (x <= 1.6):
		tmp = x / (x + (-1.0 / x))
	else:
		tmp = x + (1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.75) || !(x <= 1.6))
		tmp = Float64(x / Float64(x + Float64(-1.0 / x)));
	else
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.75) || ~((x <= 1.6)))
		tmp = x / (x + (-1.0 / x));
	else
		tmp = x + (1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.75], N[Not[LessEqual[x, 1.6]], $MachinePrecision]], N[(x / N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.75 or 1.6000000000000001 < x

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{1}{x - 1}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{x}{\color{blue}{1 + x}} + \frac{1}{x - 1} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{x + \color{blue}{-1}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{1}{x + -1} + \frac{x}{1 + x}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{1}{\frac{1 + x}{x}}} \]

      3. frac-add 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + x}{x} + \left(x + -1\right) \cdot 1}{\left(x + -1\right) \cdot \frac{1 + x}{x}}} \]

      4. *-un-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\frac{1 + x}{x}} + \left(x + -1\right) \cdot 1}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\frac{\color{blue}{x + 1}}{x} + \left(x + -1\right) \cdot 1}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

      6. *-commutative 100.0%

         \[\leadsto \frac{\frac{x + 1}{x} + \color{blue}{1 \cdot \left(x + -1\right)}}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

      7. *-un-lft-identity 100.0%

         \[\leadsto \frac{\frac{x + 1}{x} + \color{blue}{\left(x + -1\right)}}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \frac{\color{blue}{x + 1}}{x}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x + 1}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \frac{x + 1}{x}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \frac{x + 1}{x}}} \]

      2. associate-*r/ 46.5%

         \[\leadsto 1 \cdot \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\color{blue}{\frac{\left(x + -1\right) \cdot \left(x + 1\right)}{x}}} \]

      3. associate-/r/ 46.4%

         \[\leadsto 1 \cdot \color{blue}{\left(\frac{\frac{x + 1}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot x\right)} \]

      4. +-commutative 46.4%

         \[\leadsto 1 \cdot \left(\frac{\color{blue}{\left(x + -1\right) + \frac{x + 1}{x}}}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot x\right) \]

      5. associate-+l+ 46.4%

         \[\leadsto 1 \cdot \left(\frac{\color{blue}{x + \left(-1 + \frac{x + 1}{x}\right)}}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot x\right) \]

      6. *-commutative 46.4%

         \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\color{blue}{\left(x + 1\right) \cdot \left(x + -1\right)}} \cdot x\right) \]

      7. metadata-eval 46.4%

         \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\left(x + 1\right) \cdot \left(x + \color{blue}{\left(-1\right)}\right)} \cdot x\right) \]

      8. sub-neg 46.4%

         \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\left(x + 1\right) \cdot \color{blue}{\left(x - 1\right)}} \cdot x\right) \]

      9. difference-of-sqr-1 46.4%

         \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\color{blue}{x \cdot x - 1}} \cdot x\right) \]

      10. fmm-def 46.4%

          \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \cdot x\right) \]

      11. metadata-eval 46.4%

          \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \cdot x\right) \]

   8. Applied egg-rr 46.4%

      \[\leadsto \color{blue}{1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\mathsf{fma}\left(x, x, -1\right)} \cdot x\right)} \]
   9. Step-by-step derivation

      1. associate-/r/ 46.6%

         \[\leadsto 1 \cdot \color{blue}{\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}}} \]

      2. associate-/l* 46.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x + \left(-1 + \frac{x + 1}{x}\right)\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}}} \]

      3. *-lft-identity 46.6%

         \[\leadsto \frac{\color{blue}{x + \left(-1 + \frac{x + 1}{x}\right)}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      4. *-lft-identity 46.6%

         \[\leadsto \frac{x + \left(-1 + \frac{\color{blue}{1 \cdot \left(x + 1\right)}}{x}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      5. associate-*l/ 46.6%

         \[\leadsto \frac{x + \left(-1 + \color{blue}{\frac{1}{x} \cdot \left(x + 1\right)}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      6. distribute-lft-in 46.6%

         \[\leadsto \frac{x + \left(-1 + \color{blue}{\left(\frac{1}{x} \cdot x + \frac{1}{x} \cdot 1\right)}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      7. lft-mult-inverse 46.6%

         \[\leadsto \frac{x + \left(-1 + \left(\color{blue}{1} + \frac{1}{x} \cdot 1\right)\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      8. *-rgt-identity 46.6%

         \[\leadsto \frac{x + \left(-1 + \left(1 + \color{blue}{\frac{1}{x}}\right)\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      9. associate-+r+ 46.6%

         \[\leadsto \frac{x + \color{blue}{\left(\left(-1 + 1\right) + \frac{1}{x}\right)}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      10. metadata-eval 46.6%

          \[\leadsto \frac{x + \left(\color{blue}{0} + \frac{1}{x}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      11. metadata-eval 46.6%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\frac{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}{x}} \]

      12. fmm-def 46.6%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\frac{\color{blue}{x \cdot x - 1}}{x}} \]

      13. unpow2 46.6%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\frac{\color{blue}{{x}^{2}} - 1}{x}} \]

      14. div-sub 46.5%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\color{blue}{\frac{{x}^{2}}{x} - \frac{1}{x}}} \]

      15. sub-neg 46.5%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\color{blue}{\frac{{x}^{2}}{x} + \left(-\frac{1}{x}\right)}} \]

      16. unpow2 46.5%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\frac{\color{blue}{x \cdot x}}{x} + \left(-\frac{1}{x}\right)} \]

      17. associate-/l* 100.0%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\color{blue}{x \cdot \frac{x}{x}} + \left(-\frac{1}{x}\right)} \]

      18. *-inverses 100.0%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{x \cdot \color{blue}{1} + \left(-\frac{1}{x}\right)} \]

      19. *-rgt-identity 100.0%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\color{blue}{x} + \left(-\frac{1}{x}\right)} \]

      20. distribute-neg-frac 100.0%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{x + \color{blue}{\frac{-1}{x}}} \]

      21. metadata-eval 100.0%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{x + \frac{\color{blue}{-1}}{x}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{x + \left(0 + \frac{1}{x}\right)}{x + \frac{-1}{x}}} \]

   11. Taylor expanded in x around inf 98.3%

       \[\leadsto \frac{\color{blue}{x}}{x + \frac{-1}{x}} \]

   if -0.75 < x < 1.6000000000000001

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{1}{x - 1}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{x}{\color{blue}{1 + x}} + \frac{1}{x - 1} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{x + \color{blue}{-1}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{x} + \frac{1}{x + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.75 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.55:\\ \;\;\;\;\frac{x}{1 + x} + \frac{1}{x}\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.55)
   (+ (/ x (+ 1.0 x)) (/ 1.0 x))
   (if (<= x 1.6) (+ x (/ 1.0 (+ x -1.0))) (/ x (+ x (/ -1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.55) {
		tmp = (x / (1.0 + x)) + (1.0 / x);
	} else if (x <= 1.6) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = x / (x + (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.55d0)) then
        tmp = (x / (1.0d0 + x)) + (1.0d0 / x)
    else if (x <= 1.6d0) then
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    else
        tmp = x / (x + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.55) {
		tmp = (x / (1.0 + x)) + (1.0 / x);
	} else if (x <= 1.6) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = x / (x + (-1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.55:
		tmp = (x / (1.0 + x)) + (1.0 / x)
	elif x <= 1.6:
		tmp = x + (1.0 / (x + -1.0))
	else:
		tmp = x / (x + (-1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.55)
		tmp = Float64(Float64(x / Float64(1.0 + x)) + Float64(1.0 / x));
	elseif (x <= 1.6)
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = Float64(x / Float64(x + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.55)
		tmp = (x / (1.0 + x)) + (1.0 / x);
	elseif (x <= 1.6)
		tmp = x + (1.0 / (x + -1.0));
	else
		tmp = x / (x + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.55], N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.55000000000000004

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{1}{x - 1}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{x}{\color{blue}{1 + x}} + \frac{1}{x - 1} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{x + \color{blue}{-1}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.6%

      \[\leadsto \frac{x}{1 + x} + \color{blue}{\frac{1}{x}} \]

   if -0.55000000000000004 < x < 1.6000000000000001

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{1}{x - 1}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{x}{\color{blue}{1 + x}} + \frac{1}{x - 1} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{x + \color{blue}{-1}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{x} + \frac{1}{x + -1} \]

   if 1.6000000000000001 < x

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{1}{x - 1}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{x}{\color{blue}{1 + x}} + \frac{1}{x - 1} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{x + \color{blue}{-1}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{1}{x + -1} + \frac{x}{1 + x}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{1}{\frac{1 + x}{x}}} \]

      3. frac-add 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + x}{x} + \left(x + -1\right) \cdot 1}{\left(x + -1\right) \cdot \frac{1 + x}{x}}} \]

      4. *-un-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\frac{1 + x}{x}} + \left(x + -1\right) \cdot 1}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\frac{\color{blue}{x + 1}}{x} + \left(x + -1\right) \cdot 1}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

      6. *-commutative 100.0%

         \[\leadsto \frac{\frac{x + 1}{x} + \color{blue}{1 \cdot \left(x + -1\right)}}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

      7. *-un-lft-identity 100.0%

         \[\leadsto \frac{\frac{x + 1}{x} + \color{blue}{\left(x + -1\right)}}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \frac{\color{blue}{x + 1}}{x}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x + 1}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \frac{x + 1}{x}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \frac{x + 1}{x}}} \]

      2. associate-*r/ 44.4%

         \[\leadsto 1 \cdot \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\color{blue}{\frac{\left(x + -1\right) \cdot \left(x + 1\right)}{x}}} \]

      3. associate-/r/ 44.3%

         \[\leadsto 1 \cdot \color{blue}{\left(\frac{\frac{x + 1}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot x\right)} \]

      4. +-commutative 44.3%

         \[\leadsto 1 \cdot \left(\frac{\color{blue}{\left(x + -1\right) + \frac{x + 1}{x}}}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot x\right) \]

      5. associate-+l+ 44.3%

         \[\leadsto 1 \cdot \left(\frac{\color{blue}{x + \left(-1 + \frac{x + 1}{x}\right)}}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot x\right) \]

      6. *-commutative 44.3%

         \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\color{blue}{\left(x + 1\right) \cdot \left(x + -1\right)}} \cdot x\right) \]

      7. metadata-eval 44.3%

         \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\left(x + 1\right) \cdot \left(x + \color{blue}{\left(-1\right)}\right)} \cdot x\right) \]

      8. sub-neg 44.3%

         \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\left(x + 1\right) \cdot \color{blue}{\left(x - 1\right)}} \cdot x\right) \]

      9. difference-of-sqr-1 44.3%

         \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\color{blue}{x \cdot x - 1}} \cdot x\right) \]

      10. fmm-def 44.3%

          \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \cdot x\right) \]

      11. metadata-eval 44.3%

          \[\leadsto 1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \cdot x\right) \]

   8. Applied egg-rr 44.3%

      \[\leadsto \color{blue}{1 \cdot \left(\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\mathsf{fma}\left(x, x, -1\right)} \cdot x\right)} \]
   9. Step-by-step derivation

      1. associate-/r/ 44.4%

         \[\leadsto 1 \cdot \color{blue}{\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}}} \]

      2. associate-/l* 44.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x + \left(-1 + \frac{x + 1}{x}\right)\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}}} \]

      3. *-lft-identity 44.4%

         \[\leadsto \frac{\color{blue}{x + \left(-1 + \frac{x + 1}{x}\right)}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      4. *-lft-identity 44.4%

         \[\leadsto \frac{x + \left(-1 + \frac{\color{blue}{1 \cdot \left(x + 1\right)}}{x}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      5. associate-*l/ 44.4%

         \[\leadsto \frac{x + \left(-1 + \color{blue}{\frac{1}{x} \cdot \left(x + 1\right)}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      6. distribute-lft-in 44.4%

         \[\leadsto \frac{x + \left(-1 + \color{blue}{\left(\frac{1}{x} \cdot x + \frac{1}{x} \cdot 1\right)}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      7. lft-mult-inverse 44.4%

         \[\leadsto \frac{x + \left(-1 + \left(\color{blue}{1} + \frac{1}{x} \cdot 1\right)\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      8. *-rgt-identity 44.4%

         \[\leadsto \frac{x + \left(-1 + \left(1 + \color{blue}{\frac{1}{x}}\right)\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      9. associate-+r+ 44.4%

         \[\leadsto \frac{x + \color{blue}{\left(\left(-1 + 1\right) + \frac{1}{x}\right)}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      10. metadata-eval 44.4%

          \[\leadsto \frac{x + \left(\color{blue}{0} + \frac{1}{x}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

      11. metadata-eval 44.4%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\frac{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}{x}} \]

      12. fmm-def 44.4%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\frac{\color{blue}{x \cdot x - 1}}{x}} \]

      13. unpow2 44.4%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\frac{\color{blue}{{x}^{2}} - 1}{x}} \]

      14. div-sub 44.4%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\color{blue}{\frac{{x}^{2}}{x} - \frac{1}{x}}} \]

      15. sub-neg 44.4%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\color{blue}{\frac{{x}^{2}}{x} + \left(-\frac{1}{x}\right)}} \]

      16. unpow2 44.4%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\frac{\color{blue}{x \cdot x}}{x} + \left(-\frac{1}{x}\right)} \]

      17. associate-/l* 100.0%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\color{blue}{x \cdot \frac{x}{x}} + \left(-\frac{1}{x}\right)} \]

      18. *-inverses 100.0%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{x \cdot \color{blue}{1} + \left(-\frac{1}{x}\right)} \]

      19. *-rgt-identity 100.0%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{\color{blue}{x} + \left(-\frac{1}{x}\right)} \]

      20. distribute-neg-frac 100.0%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{x + \color{blue}{\frac{-1}{x}}} \]

      21. metadata-eval 100.0%

          \[\leadsto \frac{x + \left(0 + \frac{1}{x}\right)}{x + \frac{\color{blue}{-1}}{x}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{x + \left(0 + \frac{1}{x}\right)}{x + \frac{-1}{x}}} \]

   11. Taylor expanded in x around inf 99.0%

       \[\leadsto \frac{\color{blue}{x}}{x + \frac{-1}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.95:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.95) (+ x (/ 1.0 (+ x -1.0))) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.95) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.95d0) then
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.95) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.95:
		tmp = x + (1.0 / (x + -1.0))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.95)
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.95)
		tmp = x + (1.0 / (x + -1.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.95], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.94999999999999996 < x

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{1}{x - 1}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{x}{\color{blue}{1 + x}} + \frac{1}{x - 1} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{x + \color{blue}{-1}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.2%

      \[\leadsto \color{blue}{1} \]

   if -1 < x < 1.94999999999999996

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{1}{x - 1}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{x}{\color{blue}{1 + x}} + \frac{1}{x - 1} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{x + \color{blue}{-1}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{x} + \frac{1}{x + -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.0) 1.0 (if (<= x 1.0) -1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.0:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{1}{x - 1}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{x}{\color{blue}{1 + x}} + \frac{1}{x - 1} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{x + \color{blue}{-1}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.2%

      \[\leadsto \color{blue}{1} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{1}{x - 1}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{x}{\color{blue}{1 + x}} + \frac{1}{x - 1} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{x}{1 + x} + \frac{1}{x + \color{blue}{-1}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 51.0% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -1.0d0
end function

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 100.0%

   \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
2. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{1}{x - 1}} \]

   2. +-commutative 100.0%

      \[\leadsto \frac{x}{\color{blue}{1 + x}} + \frac{1}{x - 1} \]

   3. sub-neg 100.0%

      \[\leadsto \frac{x}{1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}} \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{x}{1 + x} + \frac{1}{x + \color{blue}{-1}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 50.2%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Alternative 7: 10.3% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 100.0%

   \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
2. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{1}{x - 1}} \]

   2. +-commutative 100.0%

      \[\leadsto \frac{x}{\color{blue}{1 + x}} + \frac{1}{x - 1} \]

   3. sub-neg 100.0%

      \[\leadsto \frac{x}{1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}} \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{x}{1 + x} + \frac{1}{x + \color{blue}{-1}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\frac{1}{x + -1} + \frac{x}{1 + x}} \]

   2. clear-num 100.0%

      \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{1}{\frac{1 + x}{x}}} \]

   3. frac-add 100.0%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{1 + x}{x} + \left(x + -1\right) \cdot 1}{\left(x + -1\right) \cdot \frac{1 + x}{x}}} \]

   4. *-un-lft-identity 100.0%

      \[\leadsto \frac{\color{blue}{\frac{1 + x}{x}} + \left(x + -1\right) \cdot 1}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

   5. +-commutative 100.0%

      \[\leadsto \frac{\frac{\color{blue}{x + 1}}{x} + \left(x + -1\right) \cdot 1}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

   6. *-commutative 100.0%

      \[\leadsto \frac{\frac{x + 1}{x} + \color{blue}{1 \cdot \left(x + -1\right)}}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

   7. *-un-lft-identity 100.0%

      \[\leadsto \frac{\frac{x + 1}{x} + \color{blue}{\left(x + -1\right)}}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]

   8. +-commutative 100.0%

      \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \frac{\color{blue}{x + 1}}{x}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\frac{x + 1}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \frac{x + 1}{x}}} \]

7. Taylor expanded in x around 0 73.5%

   \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\color{blue}{\frac{{x}^{2} - 1}{x}}} \]
8. Step-by-step derivation

   1. div-sub 73.5%

      \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\color{blue}{\frac{{x}^{2}}{x} - \frac{1}{x}}} \]

   2. sub-neg 73.5%

      \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\color{blue}{\frac{{x}^{2}}{x} + \left(-\frac{1}{x}\right)}} \]

   3. unpow2 73.5%

      \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\frac{\color{blue}{x \cdot x}}{x} + \left(-\frac{1}{x}\right)} \]

   4. associate-/l* 100.0%

      \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\color{blue}{x \cdot \frac{x}{x}} + \left(-\frac{1}{x}\right)} \]

   5. *-inverses 100.0%

      \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{x \cdot \color{blue}{1} + \left(-\frac{1}{x}\right)} \]

   6. *-rgt-identity 100.0%

      \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\color{blue}{x} + \left(-\frac{1}{x}\right)} \]

   7. distribute-neg-frac 100.0%

      \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{x + \color{blue}{\frac{-1}{x}}} \]

   8. metadata-eval 100.0%

      \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{x + \frac{\color{blue}{-1}}{x}} \]

9. Simplified 100.0%

   \[\leadsto \frac{\frac{x + 1}{x} + \left(x + -1\right)}{\color{blue}{x + \frac{-1}{x}}} \]
10. Step-by-step derivation

    1. associate-+r+ 100.0%

       \[\leadsto \frac{\color{blue}{\left(\frac{x + 1}{x} + x\right) + -1}}{x + \frac{-1}{x}} \]

    2. flip-+ 49.2%

       \[\leadsto \frac{\color{blue}{\frac{\left(\frac{x + 1}{x} + x\right) \cdot \left(\frac{x + 1}{x} + x\right) - -1 \cdot -1}{\left(\frac{x + 1}{x} + x\right) - -1}}}{x + \frac{-1}{x}} \]

    3. +-commutative 49.2%

       \[\leadsto \frac{\frac{\color{blue}{\left(x + \frac{x + 1}{x}\right)} \cdot \left(\frac{x + 1}{x} + x\right) - -1 \cdot -1}{\left(\frac{x + 1}{x} + x\right) - -1}}{x + \frac{-1}{x}} \]

    4. +-commutative 49.2%

       \[\leadsto \frac{\frac{\left(x + \frac{x + 1}{x}\right) \cdot \color{blue}{\left(x + \frac{x + 1}{x}\right)} - -1 \cdot -1}{\left(\frac{x + 1}{x} + x\right) - -1}}{x + \frac{-1}{x}} \]

    5. metadata-eval 49.2%

       \[\leadsto \frac{\frac{\left(x + \frac{x + 1}{x}\right) \cdot \left(x + \frac{x + 1}{x}\right) - \color{blue}{1}}{\left(\frac{x + 1}{x} + x\right) - -1}}{x + \frac{-1}{x}} \]

    6. +-commutative 49.2%

       \[\leadsto \frac{\frac{\left(x + \frac{x + 1}{x}\right) \cdot \left(x + \frac{x + 1}{x}\right) - 1}{\color{blue}{\left(x + \frac{x + 1}{x}\right)} - -1}}{x + \frac{-1}{x}} \]

11. Applied egg-rr 49.2%

    \[\leadsto \frac{\color{blue}{\frac{\left(x + \frac{x + 1}{x}\right) \cdot \left(x + \frac{x + 1}{x}\right) - 1}{\left(x + \frac{x + 1}{x}\right) - -1}}}{x + \frac{-1}{x}} \]
12. Step-by-step derivation

    1. difference-of-sqr-1 49.2%

       \[\leadsto \frac{\frac{\color{blue}{\left(\left(x + \frac{x + 1}{x}\right) + 1\right) \cdot \left(\left(x + \frac{x + 1}{x}\right) - 1\right)}}{\left(x + \frac{x + 1}{x}\right) - -1}}{x + \frac{-1}{x}} \]

    2. metadata-eval 49.2%

       \[\leadsto \frac{\frac{\left(\left(x + \frac{x + 1}{x}\right) + \color{blue}{\left(--1\right)}\right) \cdot \left(\left(x + \frac{x + 1}{x}\right) - 1\right)}{\left(x + \frac{x + 1}{x}\right) - -1}}{x + \frac{-1}{x}} \]

    3. sub-neg 49.2%

       \[\leadsto \frac{\frac{\color{blue}{\left(\left(x + \frac{x + 1}{x}\right) - -1\right)} \cdot \left(\left(x + \frac{x + 1}{x}\right) - 1\right)}{\left(x + \frac{x + 1}{x}\right) - -1}}{x + \frac{-1}{x}} \]

    4. sub-neg 49.2%

       \[\leadsto \frac{\frac{\left(\left(x + \frac{x + 1}{x}\right) - -1\right) \cdot \color{blue}{\left(\left(x + \frac{x + 1}{x}\right) + \left(-1\right)\right)}}{\left(x + \frac{x + 1}{x}\right) - -1}}{x + \frac{-1}{x}} \]

    5. metadata-eval 49.2%

       \[\leadsto \frac{\frac{\left(\left(x + \frac{x + 1}{x}\right) - -1\right) \cdot \left(\left(x + \frac{x + 1}{x}\right) + \color{blue}{-1}\right)}{\left(x + \frac{x + 1}{x}\right) - -1}}{x + \frac{-1}{x}} \]

    6. associate-+r+ 49.2%

       \[\leadsto \frac{\frac{\left(\left(x + \frac{x + 1}{x}\right) - -1\right) \cdot \color{blue}{\left(x + \left(\frac{x + 1}{x} + -1\right)\right)}}{\left(x + \frac{x + 1}{x}\right) - -1}}{x + \frac{-1}{x}} \]

    7. +-commutative 49.2%

       \[\leadsto \frac{\frac{\left(\left(x + \frac{x + 1}{x}\right) - -1\right) \cdot \left(x + \color{blue}{\left(-1 + \frac{x + 1}{x}\right)}\right)}{\left(x + \frac{x + 1}{x}\right) - -1}}{x + \frac{-1}{x}} \]

    8. associate-/l* 100.0%

       \[\leadsto \frac{\color{blue}{\left(\left(x + \frac{x + 1}{x}\right) - -1\right) \cdot \frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\left(x + \frac{x + 1}{x}\right) - -1}}}{x + \frac{-1}{x}} \]

13. Simplified 100.0%

    \[\leadsto \frac{\color{blue}{\left(x + \left(2 + \frac{1}{x}\right)\right) \cdot \frac{x + \left(0 + \frac{1}{x}\right)}{x + \left(2 + \frac{1}{x}\right)}}}{x + \frac{-1}{x}} \]

15. Applied egg-rr 10.2%

    \[\leadsto \color{blue}{-2} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))